Computer Science > Logic in Computer Science

Abstract: Principia Logico-Metaphysica proposes a foundational logical theory for
metaphysics, mathematics, and the sciences. It contains a canonical development
of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of
Ernst Mally, formalized by Zalta) that differentiates between ordinary and
abstract objects.
This article reports on recent work in which AOT has been successfully
represented and partly automated in the proof assistant system Isabelle/HOL.
Initial experiments within this framework reveal a crucial but overlooked fact:
a deeply-rooted and known paradox is reintroduced in AOT when the logic of
complex terms is simply adjoined to AOT's specially-formulated comprehension
principle for relations. This result constitutes a new and important paradox,
given how much expressive and analytic power is contributed by having the two
kinds of complex terms in the system. Its discovery is the highlight of our
joint project and provides strong evidence for a new kind of scientific
practice in philosophy, namely, computational metaphysics.
Our results were made technically possible by a suitable adaptation of
Benzm\"uller's metalogical approach to universal reasoning by semantically
embedding theories in classical higher-order logic. This approach enables the
fruitful reuse of state-of-the-art higher-order proof assistants, such as
Isabelle/HOL, for mechanizing and experimentally exploring challenging logics
and theories such as AOT. Our results also provide a fresh perspective on the
question of whether relational type theory or functional type theory better
serves as a foundation for logic and metaphysics.